Place sweetbreads in a medium mixing bowl; cover with water. Set aside to soak for 1 hour. Drain well; remove and discard white membrane.
Combine sweetbreads, water, celery leaves, onion, 1 teaspoon salt, and 1/8 teaspoon pepper in a medium Dutch oven. Bring to a boil. Cover; reduce heat, and simmer 30 minutes.
Remove sweetbreads, and cut into serving-size pieces; set aside. Strain broth, and reserve 1 cup. Discard vegetables and remaining broth.
Combine flour and remaining salt and pepper. Dredge sweetbreads in flour; set aside.
Melt butter in a skillet over medium heat. Add sweetbreads; cook over low heat 10 minutes or until browned, turning as necessary. Remove sweetbreads; drain. Reserve pan drippings. Transfer sweetbreads to a warm serving platter.
Pour reserved 1 cup broth into pan drippings. Cook over medium heat, stirring frequently, until thickened and bubbly. Serve gravy with sweetbreads.